Citation pilot reported a NMAC on initial climb with an air carrier departing an adjacent airport. Citation flight crew took evasive action to avoid a possible collision.

Narrative

We were given a delay holding short of the runway at ZZZ; an airport about 10 miles west of ZZZ1. Holding short of the runway; ATC told Myself and the other pilot about a POTENTIAL 10 minute delay due to departure congestion at ZZZ1 and they had to stop departures for us to takeoff. After about 10 min we were given a takeoff clearance in a Complex large Citation Jet... Off Runway XXR HDG 345 Maintain 3000ft; Which we complied with. Once being switched to ZZZ Departure we were given an instruction to turn to HDG 120 and to maintain 3000ft for traffic. 15 seconds later we see an aircraft on our TCAS airborne out of ZZZ1 off Runway XYR flying in our general direction. ATC about 5 seconds later says; Aircraft Y are you up? Shortly after the pilot checks on and says yes; 2000ft climbing 5000 ft at this moment we get a 'TRAFFIC TRAFFIC' ON our TCAS. ATC tells the Aircraft Y plane to turn left HDG 180. Immediately when the pilot responds to ' hdg 180 Aircraft Y'; about 5 seconds after this occurs my co-pilot says '[Expletive] they are turning right into us'. I'm the Pilot flying and am hand flying the aircraft... I hear Descend NOW Descend Now and respond to the RA by following the Proc we are trained to put the flight director in the green RA box and follow it (Hard left turn and a aggressive descent) at this point ATC gives us a immediate traffic alert and a HDG to the left and a Climb. We tell ATC we are Complying with an RA standby. The matter resolves itself with the Aircraft Y eventually turning to a left hdg 180. The Aircraft Y made a right turn HDG 180 because we got the RA. The RA did RESULT in my aircraft in an Low altitude situation briefly but by the time we were clear of the traffic we were visual outside and could maintain terrain avoidance. ATC ZZZ dep starts saying on frequency how ZZZ1 Tower should not have launched the Aircraft Y as that put that plane right into our path and they are figuring out right now what happened; Also the Aircraft Y pilot was told he turned the wrong way again into us and causing a traffic conflict. The Aircraft Y pilot was given a Possible pilot deviation. Myself and co captain did nothing wrong and flew the plane how we are trained in a traffic conflict situation. My takeaways from this event are: A need for better coordination between ZZZ1; ZZZ [TRACON]; and ZZZ for IFR DEP. Better procedures to get airplanes safely out of the airspace with so many airports so closely spaced together. Continued pilot training annually on traffic conflict events.